An elementary proof of the Birkhoff-Hopf theorem

Birkhoff used what has been called ([7]) 'Hilbert's projective metric' or ([9]) the 'Cayley-Hilbert metric'. In each case, it proved possible to obtain sharp estimates for the contraction constant of a positive linear operator with respect to the ' almost' metric. Subsequently, several authors generalized and sharpened the original results and established a close connection between the Birkhoff and Hopf theorems. A partial list of contributors includes F. L. Bauer[l], M. A. Ostrowski[25, 26] and P. J. Bushell [8, 7, 9]. In addition, a number of mathematicians who were apparently unaware of most of the above-mentioned theorems obtained closely related results and interesting new propositions. We mention A. M. Krasnosel'skii, Je. E. Lifshits, Yu. V. Pokornyi, A. V. Sobolev, and refer the reader to [19], [31] and the book [18]. We shall prove here a generalization of the work of Birkhoff, Hopf, Bauer, Ostrowski, Bushell and others and refer to the cumulative result as the Birkhoff-Hopf Theorem: see Theorems 3-5 and 3-6 below and the formulae of Section 6. Typically, when the Birkhoff-Hopf Theorem is applied to a positive (possibly noncompact) linear operator L, it implies that the L has a unique, normalized, positive eigenvector v with corresponding eigenvalue A equal to the spectral radius of L and that there are explicitly computable constants M and c, with c < 1, such that